Simulations
We performed two sets of simulations. In the first set, we simulated timecourses of temperature rises for all three probes following a heat pulse, for sap velocity ranging from -10 to 80 cm h-1. We added Gaussian noise with a standard deviation of 0.02 K (using simulation module randn in MATLAB) to each timecourse. The code and procedures for generating the synthetic data in each timecourse and estimated sap velocity by each of the methods listed above are provided in the SI: Methods S1 . Each such simulation was repeated 103 times to estimate the probability distribution of each estimate of sap flux under random temperature noise. All simulations used sapwood properties given in TableS1 .
In the second set of simulations (Methods S2 ), we determined the sensitivity of each method to inaccuracy in k . Although heat pulse methods typically assume constant k , in practice it varies with mc , which may vary diurnally or seasonally in relation to changes in water potential and cycles of discharge and recharge of trunk water stores (Chen et al., 2012; López-Bernal et al., 2014). We simulated timecourses of temperature rises as for the first set of simulations for sap velocities between -10 to 80 cm h-1, while assuming each of three mc = 0.5, 1.0, 1.5 g g-1. We calculated k from mcfor a basic density (ρ b) of 0.5·103 kg m-3 using the relationship given by Vandegehuchte & Steppe (2012b) (see Fig.S2 ).
2.2.2 Theoretical test of internal calibration of k
Because k may vary in relation to fluctuations in mc , using a constant value of k in the DRM (or HRM) may cause systematic errors. To reduce such errors, k can be estimatedin vivo by combining the sap velocity estimate from the DRM with an independent estimate of V , obtained from the CHPM, which does not explicitly depend on k . Although the CHPM itself has limitations (see Introduction), this procedure provides useful information about the magnitude of diurnal and seasonal changes ink . Setting V DRM equal toV CHPM under conditions of high sap velocity (so that V DRM = V 23) and solving for k gives